#7498678

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2,4,5)(3,6,7,8)(10,14)(11,17)(12,20)(22,24)(25,26,32,45,31,37,30,36)(27,33,46,55,40,50,39,49)(28,41,42,34,29,35,43,51)(38,48,53,47,44,52,54,56);(2,5)(3,7)(9,10,13,14)(11,18,17,15)(12,21,20,16)(19,22,23,24)(25,27)(26,33)(28,38)(29,44)(30,39)(31,40)(32,46)(34,47)(35,52)(36,49)(37,50)(41,48)(42,53)(43,54)(45,55)(51,56);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28,31,29)(26,34,37,51)(27,38,40,44)(30,42,32,43)(33,47,50,56)(35,45,41,36)(39,53,46,54)(48,49,52,55);(1,3,4,7)(2,6,5,8)(11,17)(15,18)(19,23)(22,24)(25,29,31,28)(26,35,37,41)(27,38,40,44)(30,42,32,43)(33,48,50,52)(34,45,51,36)(39,54,46,53)(47,49,56,55);(2,5)(6,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,30,31,32)(26,36,37,45)(27,39,40,46)(28,42,29,43)(33,49,50,55)(34,35,51,41)(38,53,44,54)(47,52,56,48)

Polycyclic presentation:

2^9|a^2=fgi,b^2=h,c^2=i,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7405874);